Annihilation creation operator

In summary, the conversation is about calculating the expectation value of the position x using the wave function psi(x) = |1> + |2> and the equation <x> = <psi|(a+a^+|psi>, with a being the annihilation operator and a+ being the creation operator. The goal is to find the expectation value by expanding the calculation, but the results obtained using the properties of annihilation and creation operators are incorrect. It is then discussed that <1|1> and <2|2> are zero, and the states |1> and |2> are orthonormal.
  • #1
greisen
76
0
Hey,

I am calculating the expectation value of the position x. I have the wave function
psi(x) = |1> + |2>

so I use the equation <x> = <psi|(a+a^+|psi> to calculate the mean value. So I get

(<1| + <2|)(a+a^+)(|1> + |2>)

which I reduce to

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance
 
Physics news on Phys.org
  • #2
greisen said:
Hey,

I am calculating the expectation value of the position x. I have the wave function
psi(x) = |1> + |2>

so I use the equation <x> = <psi|(a+a^+|psi> to calculate the mean value. So I get

(<1| + <2|)(a+a^+)(|1> + |2>)

which I reduce to

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance
I am not following the notation. If a is an operator and a^ is an operator, then why is there a + after a^ and why are there no a^ in your expansion?
 
Last edited:
  • #3
Maybe I have expressed the problem badly. a is the annihilation operator and a+ is the creation operator. I have a two state system |1>, |2> with a wavefunction (psi) = |1> + |2>. My problem is to perform the multiplication in order to find the expectation value:

(<1| + <2|)(a + a+)(|1> + |2>)

if I expand this calculation I get some strange results. How to multiple this? Any help appreciated - thanks in advance
 
  • #4
greisen said:
Hey,

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance

<1|0>=<2|3>=0

In the above sum, only two terms are nonzero.
 
  • #5
greisen said:
Maybe I have expressed the problem badly. a is the annihilation operator and a+ is the creation operator. I have a two state system |1>, |2> with a wavefunction (psi) = |1> + |2>. My problem is to perform the multiplication in order to find the expectation value:

(<1| + <2|)(a + a+)(|1> + |2>)

if I expand this calculation I get some strange results. How to multiple this? Any help appreciated - thanks in advance
So a|n> = |n-1> and a+|n> = |n+1>? and the states are orthogonal? Is that right? Euclid seems to know what you are doing, and it seems to be consistent with this interpretation. What are <1|1> and <2|2>? Are the states orthonormal? This is not a hint. I am asking because I want to know.
 
  • #6
Yes the two kets |1> and |2> are orthonormale. My problem in the multiplication was the <2|3>, <1|0> which are zero.
 

FAQ: Annihilation creation operator

What is an annihilation creation operator?

An annihilation creation operator is a mathematical operator used in quantum mechanics to describe the creation and annihilation of particles. It is used to model the behavior of quantum systems and is an essential tool for understanding phenomena such as particle creation and annihilation.

How does an annihilation creation operator work?

An annihilation creation operator operates on a quantum state to either create or annihilate a particle. It is represented by the symbols "a" and "a†" and has specific mathematical rules for how it interacts with other quantum operators and states.

What is the purpose of an annihilation creation operator?

The purpose of an annihilation creation operator is to describe the behavior of particles in quantum systems. It allows scientists to study and predict the behavior of particles and their interactions, which is crucial for understanding the fundamental laws of physics.

What are the properties of an annihilation creation operator?

An annihilation creation operator has several properties, including linearity, Hermitian conjugation, and commutation relations. These properties are essential for understanding how the operator behaves and how it interacts with other operators in quantum systems.

How is an annihilation creation operator used in experiments?

An annihilation creation operator is used in experiments that involve the creation and annihilation of particles, such as in particle accelerators. It is also used in theoretical calculations and simulations to study the behavior of quantum systems and make predictions about their behavior.

Back
Top